Real Numbers — NCERT Class 10 Complete Lesson
?? Euclid's Division Lemma
For any two positive integers a and b, there exist unique integers q (quotient) and r (remainder) such that:
Example: For a = 455 and b = 42: 455 = 42 × 10 + 35 ? (remainder 35 < 42)
This lemma is used to find the HCF (Highest Common Factor) of two numbers.
?? The Fundamental Theorem of Arithmetic
Every composite number can be expressed as a product of primes, and this factorisation is unique (except for the order of factors).
No matter how you factorise 12, you always get the same prime factors!
Use this to find HCF and LCM:
HCF = product of smallest powers of common primes
LCM = product of greatest powers of all primes
v2 is Irrational — Proof
Assume v2 = p/q (in lowest terms). Then 2 = p²/q², so p² = 2q². This means p² is even, so p is even. Let p = 2m. Then 4m² = 2q², so q² = 2m², meaning q is also even. But then p and q have a common factor 2 — contradiction! Hence v2 is irrational.
e.g., 3 + v2 is irrational. 5v3 is irrational.
Real Numbers — MCQ Practice
Real Numbers — Mnemonic
Mnemonic 1: REAL NUMBERS Classification 📚
"Ravi's Elephant Always Loves Numbers, Usually Making Big Easy Real Stories"
- Ravi's - Rational Numbers
- Elephant - Irrational Numbers
- Always - Real Numbers
- Loves - Whole Numbers
- Numbers - Natural Numbers
- Usually - Integers
- Making - Fractions (Rational)
- Big - Perfect Squares
- Easy - Repeating Decimals
- Real - Irrational Decimals
- Stories - Zero (included in Integers)
Mnemonic 2: Remember Types of Real Numbers 🧠
"NITWIPR" (pronounced 'nit-wip-er') — Natural, Integer, Terminating decimals, Whole, Irrational, Perfect squares, Rational
Use this to recall all types quickly during exams!
Mnemonic 3: Hindi Rhyming Trick for Real Numbers 🇮🇳
"प्राकृतिक, पूर्ण, पूर्णांक, भिन्न, अपरिमेय, सब हैं रियल नंबर के रंग"
(Prakritik, Poorn, Poornank, Bhinn, Aparimey, Sab Hain Real Number Ke Rang)
- प्राकृतिक (Natural)
- पूर्ण (Whole)
- पूर्णांक (Integer)
- भिन्न (Rational/Fractions)
- अपरिमेय (Irrational)
This rhyme helps you remember all main types of real numbers in Hindi with rhythm and ease!
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